Skip to main content

The upper bound estimate of the number of integer points on elliptic curves y 2 = x 3 + p 2 r x

Abstract

Let p be a fixed prime and r be a fixed positive integer. Further let N( p 2 r ) denote the number of pairs of integer points (x,±y) on the elliptic curve E: y 2 = x 3 + p 2 r x with y>0. Using some properties of Diophantine equations, we give a sharper upper bound estimate for N( p 2 r ). That is, we prove that N( p 2 r )1, except with N( 17 2 ( 2 s + 1 ) )=2, where s is a nonnegative integer.

MSC:11G05, 11Y50.

1 Introduction

Let , be the sets of all integers and positive integers, respectively. Let p be a fixed prime and k be a fixed positive integer. Recently, the integer points (x,y) on the elliptic curve

y 2 = x 3 + p k x
(1.1)

have been investigated in many papers (see [1, 2] and [3]). In this paper we deal with the number of integer points on (1.1) for even k. Then (1.1) can be rewritten as

y 2 = x 3 + p 2 r x,
(1.2)

where r is a positive integer.

An integer point (x,y) on (1.2) is called trivial or non-trivial according to whether y=0 or not. Obviously, (1.2) has only the trivial integer point (x,y)=(0,0). Notice that if (x,y) is a non-trivial integer point on (1.2), then (x,y) is also. Therefore, (x,y) along with (x,y) are called by a pair of non-trivial integer points and denoted by (x,±y), where y>0. For any positive integer n, let

u(n)= 1 2 ( α n + β n ) ,v(n)= 1 2 2 ( α n β n ) ,
(1.3)

where

α=3+2 2 ,β=32 2 .
(1.4)

Using some properties of Diophantine equations, we give a sharper upper bound estimate for N( p 2 r ), the number of pairs of non-trivial integer points (x,±y) on (1.2). That is, we shall prove the following results.

Theorem 1.1 All non-trivial integer points on (1.2) are given as follows.

  1. (i)

    p=u( 2 m ), r=2s+1, (x,±y)=( p 2 s v 2 ( 2 m ),± p 3 s v( 2 m )( v 2 ( 2 m )+1)), where m, s are nonnegative integers.

  2. (ii)

    p1(mod8), r=2s+1, (x,±y)=( p 2 s + 1 X 2 ,± p 3 s + 2 XY), where s is a nonnegative integer, (X,Y) is a solution of the equation

    X 4 p Y 2 =1,X,YN.
    (1.5)

Theorem 1.2 Let p be an odd prime, r be a positive integer. Then for any nonnegative integer s, we have N( p 2 r )1, except with N( 17 2 ( 2 s + 1 ) )=2. Moreover, if p1(mod8), then N( p 2 r )=0, except with N( 3 2 ( 2 s + 1 ) )=1.

2 Lemmas

Lemma 2.1 ([[4], Theorem 244])

Every solution (u,v) of the equation

u 2 2 v 2 =1,u,vN
(2.1)

can be expressed as (u,v)=(u(n),v(n)), where n is a positive integer.

Lemma 2.2 If p=u(n), then n= 2 m , where m is a nonnegative integer.

Proof Assume that n has an odd divisor d with d>1. Then we have either u(1)|u(n) and 1<u(1)<u(n) or u(n/d)|u(n) and 1<u(n/d)<u(n). Therefore, since p is a prime, it is impossible. Thus, we get n= 2 m . The lemma is proved. □

Any fixed positive integer a can be uniquely expressed as a=b c 2 , where b, c are positive integers with b is square free. Then b is called the quadratfrei of a and denoted by Q(a).

Lemma 2.3 For any positive integer m, we have 3|Q(v( 2 m )).

Proof By (1.3) and (1.4), we get

v ( 2 m ) = 2 m + 1 i = 0 m 1 u ( 2 i )
(2.2)

and

u ( 2 i ) =2 u 2 ( 2 i 1 ) 1,iN.
(2.3)

Since (2/3)=1, where (2/3) is the Legendre symbol, we see from (2.3) that 3u( 2 i ) for i1. Therefore, since u(1)=3, by (2.2), we obtain 3v( 2 m ). It implies that 3|Q(v( 2 m )). The lemma is proved. □

Let D be a non-square positive integer. It is a well known fact that if the equation

U 2 D V 2 =1,U,VN
(2.4)

has solutions (U,V), then it has a unique solution ( U 1 , V 1 ) such that U 1 + V 1 D U+V D , where (U,V) through all solutions of (2.4). For any odd positive integer l, let

U(l)+V(l) D = ( U 1 + V 1 D ) l .

Then (U,V)=(U(l),V(l)) (l=1,3,) are all solutions of (2.4).

Lemma 2.4 ([[5], Theorem 1])

The equation

X 4 D Y 2 =1,X,YN
(2.5)

has at most one solution (X,Y). Moreover, if the solution (X,Y) exists, then ( X 2 ,Y)=(U(l),V(l)), where l=Q( U 1 ).

Lemma 2.5 ([[5], Theorem 3])

If 3|Q( U 1 ), then (2.5) has no solutions (X,Y).

Lemma 2.6 If p=u( 2 m ), where m is a positive integer with m>1, then (1.5) has no solutions (X,Y).

Proof Since p=u( 2 m ) with m>1, by (2.3), we have

p=2 u 2 ( 2 m 1 ) 1=4 v 2 ( 2 m 1 ) +1.
(2.6)

We see from (2.6) that the equation

U 2 p V 2 =1,U,VN
(2.7)

has solution (U,V) and its fundamental solution is ( U 1 , V 1 )=(2v( 2 m 1 ),1). Further, since m11, by Lemma 2.3, we have 3|Q(v( 2 m 1 )). Hence, we get 3|Q( U 1 )=Q(2v( 2 m 1 )). Therefore, by Lemma 2.5, the lemma is proved. □

Lemma 2.7 ([6])

The equation

2 X 2 +1= Y n ,X,Y,nN,n>3
(2.8)

has no solutions (X,Y,n).

Lemma 2.8 The equation

X 2 Y 4 = p 2 n ,X,Y,nN,gcd(X,Y)=1
(2.9)

has only the solutions (p,X,Y,n)=(u( 2 m ), v 2 ( 2 m )+1,v( 2 m ),1), where m is a nonnegative integer.

Proof Assume that (p,X,Y,n) is a solution of (2.9). If p=2, since gcd(X,Y)=1, then we have 2XY, gcd(X+ Y 2 ,X Y 2 )=2, X+ Y 2 = 2 2 n 1 , X Y 2 =2 and Y 2 = 2 2 n 2 1. But since Y 2 +1 is not a square, it is impossible.

If p is an odd prime, then we have gcd(X+ Y 2 ,X Y 2 )=1, and by (2.9), we get X+ Y 2 = p 2 n , X Y 2 =1,

2X= p 2 n +1
(2.10)

and

2 Y 2 = p 2 n 1.
(2.11)

By Lemma 2.7, we get from (2.11) that n=1 and

p 2 2 Y 2 =1.
(2.12)

Further, applying Lemma 2.1 to (2.12) yields

p=u(n),Y=v(n),nN.
(2.13)

Further, by Lemma 2.2, we see from the first equality of (2.13) that n= 2 m . Thus, by (2.10) and (2.13), the lemma is proved. □

3 Proof of Theorem 1.1

Assume that (x,±y) is a pair of non-trivial integer points on (1.2). Since y>0, we have x>0 and x can be expressed as

x= p t z,tZ,t0,zN,pz.
(3.1)

Substituting (3.1) into (1.2) yields

p t z ( p 2 t z 2 + p 2 r ) = y 2 .
(3.2)

We first consider the case that r>t. By (3.2), we have

p 3 t z ( z 2 + p 2 r 2 t ) = y 2 .
(3.3)

Since pz, we have p z 2 + p 2 r 2 t and gcd(z, z 2 + p 2 k 2 r )=1. Hence by (3.3), we get

t = 2 s , z = f 2 , z 2 + p 2 r 2 t = g 2 , y = p 3 s f g , s Z , s 0 , f , g N , gcd ( f , g ) = 1 ,
(3.4)

whence we obtain

g 2 f 4 = p 2 r 4 s .
(3.5)

Applying Lemma 2.8 to (3.5) yields

p=u ( 2 m ) ,2r4s=2,f=v ( 2 m ) ,g= v 2 ( 2 m ) +1,mZ,m0.
(3.6)

Therefore, by (3.1), (3.4), and (3.6), the integer points of type (i) are given.

We next consider the case that r=t. Then we have

p 3 r z ( z 2 + 1 ) = y 2 .
(3.7)

Since pz, gcd(z, z 2 +1)=1 and z 2 +1 is not a square, we see from (3.7) that

r = 2 s + 1 , z = f 2 , z 2 + 1 = p g 2 , y = p 3 s + 2 f g , s Z , s 0 , f , g N , gcd ( f , g ) = 1 .
(3.8)

By (3.8), we get

f 4 p g 2 =1.
(3.9)

It implies that (X,Y)=(f,g) is a solution of (1.5). Therefore, by (3.1) and (3.8), we obtain the integer points of type (ii).

We finally consider the case that r<t. Then we have

p t + 2 r z ( p 2 t 2 r z 2 + 1 ) = y 2 .
(3.10)

Since pz( p 2 t 2 r z 2 +1) and gcd(z, p 2 t 2 r z 2 +1)=1, we see from (3.10) that p 2 t 2 r z 2 +1 is a square, a contradiction.

To sum up, the theorem is proved.

4 Proof of Theorem 1.2

By (2.3), if p=u( 2 m ) with m1, then p1(mod8). Therefore, by Theorem 1.1, if p1(mod8), then (1.2) has only the non-trivial integer point

p=3,r=2s+1,(x,±y)= ( 3 2 s 4 , ± 3 3 s 10 ) .
(4.1)

It implies that the theorem is true for p1(mod8).

For p1(mod8), let N 1 and N 2 denote the number of pairs of non-trivial integer points of types (i) and (ii) in Theorem 1.1, respectively. Obviously, we have

N ( p 2 r ) = N 1 + N 2
(4.2)

and N 1 1. By Lemma 2.4, we get N 2 1. Hence, by (4.2), we have N( p 2 r )2 for p1(mod8). Since u(2)=17 and (1.5) has the solution (X,Y)=(2,1) for p=17, by Theorem 1.1, we get

p = 17 , r = 2 s + 1 , ( x , ± y ) = ( 17 2 s 144 , ± 17 3 s 1 , 740 ) and ( 17 2 s + 1 4 , ± 17 3 s + 2 2 )
(4.3)

and N( 17 2 ( 2 s + 1 ) )=2. However, by Lemma 2.6, if p=u( 2 m ) with m>1, then N 2 =0. Therefore, by (4.2), if p1(mod8), then N( p 2 r )1, except with N( 17 2 ( 2 s + 1 ) )=2. The theorem is proved.

References

  1. Bremner A, Cassels JWS:On the equation Y 2 =X( X 2 +p). Math. Comput. 1984, 42: 247-264.

    MathSciNet  MATH  Google Scholar 

  2. Draziotis KA:Integer points on the curve Y 2 = X 3 ± p k X. Math. Comput. 2006, 75: 1493-1505. 10.1090/S0025-5718-06-01852-7

    Article  MathSciNet  MATH  Google Scholar 

  3. Walsh PG:Integer solutions to the equation y 2 =x( x 2 ± p k ). Rocky Mt. J. Math. 2008,38(4):1285-1302. 10.1216/RMJ-2008-38-4-1285

    Article  MathSciNet  MATH  Google Scholar 

  4. Hardy GH, Wright EM: An Introduction to the Theory of Numbers. 5th edition. Oxford University Press, Oxford; 1979.

    MATH  Google Scholar 

  5. Cohn JHE:The Diophantine equation x 4 +1=D y 2 . Math. Comput. 1997, 66: 1347-1351. 10.1090/S0025-5718-97-00851-X

    Article  MATH  Google Scholar 

  6. Nagell T: Sur l’impossibilité de quelques equations à deux indéterminées. Norsk Mat. Forenings Skr. 1921,13(1):65-82.

    Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referees for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the P.E.D. (2013JK0573) and N.S.F. (11371291) of P.R. China.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Xiaoxue Li.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

JZ obtained the theorems and completed the proof. XL corrected and improved the final version. Both authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Zhang, J., Li, X. The upper bound estimate of the number of integer points on elliptic curves y 2 = x 3 + p 2 r x. J Inequal Appl 2014, 104 (2014). https://doi.org/10.1186/1029-242X-2014-104

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2014-104

Keywords

  • elliptic curve
  • integer point
  • Diophantine equation